Solution045_121 - Problem Set 4. - Solutions J. Scholtz...

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Problem Set 4 - Solutions J. Scholtz October 23, 2006 Question 1: 2.5/8 We can pretty much follow the proof of theorem 2.23: Let T : V W and β , β 0 be the ordered bases for V and similarly with γ ’s for W . Then we can consider the transformation: [ T ] γ β Q = [ T ] γ β [ I ] β β 0 = [ T ] γ β 0 We can also check that P - 1 switches γ to γ 0 : PP - 1 = I γ , hence the result is in γ basis but the argument of P is in the γ 0 basis. Moreover: PP - 1 = [ I ] γ γ 0 P - 1 = I γ , therefore P - 1 = [ I ] γ 0 γ . And thence by repeating the argument from first part: [ T ] γ 0 β 0 = P - 1 [ T ] γ β Q Question 2: 2.5/10 If A and B are similar matrices, then there exists an invertible matrix Q , such that B = Q - 1 AQ . Then by exercise 13 in section 2.3: Tr( CD ) = Tr( DC ), hence: Tr( B ) = Tr( Q - 1 AQ ) = Tr( AQQ - 1 ) = Tr( AI ) = Tr( A ) Question 3: 2.6/4 Let V = R 3 , and let there be f 1 ,f 2 ,f 3 V * such that: f 1 = x - 2 y f 2 = x + y + z f 3 = y - 3 z Since we know that dim( V * ) = dim( V ) = 3, since V is finite dimensional, then it is enough to show span or linear independence. Let us show linear independence.
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Solution045_121 - Problem Set 4. - Solutions J. Scholtz...

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